Abstract

We describe and study an explicit structure of a regular cell complex $ \mathcal{K}(L)$
on the moduli space $ M(L)$ of a planar polygonal linkage $ L$. The
combinatorics is very much related (but not equal) to the combinatorics of the
permutohedron. In particular, the cells of maximal dimension are labeled by
elements of the symmetric group. For example, if the moduli space $ M$
is a sphere, the complex $ \mathcal{K}$ is dual to the boundary complex of the
permutohedron.The dual complex $ \mathcal{K}^*$ is patched of Cartesian products
of permutohedra. It can be explicitly realized in the Euclidean space via a
surgery on the permutohedron.

Keywords

1.
Preliminaries and Notation

A polygonal
$ n$ -linkage
is a sequence of positive numbers $ L=(l_1,\dots ,l_n)$. It should be interpreted as a
collection of rigid bars of lengths $ l_i$ joined consecutively in a chain by
revolving joints. We always assume that the triangle inequality holds, that is,
\begin{eqnarray*} \forall j, \ \ \ l_j< \frac{1}{2}\sum_{i=1}^n l_i \end{eqnarray*} which guarantees that the chain of bars can close.
A planar configuration
of $ L$ is a sequence of points \begin{eqnarray*} P=(p_1,\dots,p_{n}), \ p_i \in \mathbb{R}^2 \end{eqnarray*} with $ l_i=|p_i,p_{i+1}|$, and
$ l_n=|p_n,p_{1}|$. We also call $ P$ a
polygon
.

As follows from the definition, a configuration may have
self-intersections and/or self-overlappings.

Definition 1.1.

The moduli
space, or the configuration space
$ M(L)$ is the set of all configurations of $ L$ modulo orientation
preserving isometries of $ \mathbb{R}^2$. Equivalently, we can define
$ M(L)$ as \begin{eqnarray*} M(L)=\{(u_1,\ldots,u_n) \in (S^1)^n : \sum_{i=1}^n l_iu_i=0\}/SO(2). \end{eqnarray*}

The (second) definition shows that $ M(L)$ does not depend on the
ordering of $ \{l_1,\ldots,l_n\}$; however, it does depend on the values of $ l_i$.

Throughout the paper (except for the Sect. 4) we assume
that no configuration of $ L$ fits a straight line. This assumption
implies that the moduli space $ M(L)$ is a closed
$ (n-3)$-dimensional manifold (see [Farber2008]).

The manifold $ M(L)$ is already well studied, see [Farber2008], [Farber and Schütz2007],[Kapovich and Millson1995], and many other
papers. Explicit descriptions of $ M(L)$ exist for $ n=4, 5, {\rm and}\, 6$, see [Farber2008],[Kapovich and Millson1995],[Zvonkine1997]. There also exist various results for
polygonal linkages in 3D, see [Klyachko1994] for example.

The paper is organized as follows. Section 2 presents an
explicit combinatorial description of $ M(L)$ as a regular cell complex
$ \mathcal{K}(L)$. In a sense, the starting point of our approach is an elementary
version of Gelfand–Goresky–MacPherson–Serganova idea
from [Gelfand et al.1987]: they classify the planes (that
is, the elements of Grassmanian) by some associated combinatorics. The
equivalence classes of the planes form strata
which may have complicated topology. In this paper we also classify
configurations by their combinatorial types, but here we are lucky with that
all equivalence classes are topological balls that patch together in a regular cell
complex. The combinatorics of $ \mathcal{K}(L)$ is very much related (but not
equal) to the combinatorics of the permutohedron. In Sect. 2 we present a
number of examples and give a complete characterization of the possible
combinatorics of cells.

In Sect. 3 we study the dual complex $ \mathcal{K}^*$ which
comes almost automatically with a geometrical realization in the Euclidean
space. The realization is related to
cyclopermutohedron [Panina2015], which is a polytope that encodes
cyclically ordered partitions of a finite set in the same way as the permutohedron
encodes linearly ordered partitions.

Section 4 sketches the main result of [Galashin and Panina2016]: under a proper setting,
a "polygonal linkage" can be replaced by a "simple game" (in the
game-theoretic sense). A simple game cannot be interpreted as a physical
object (like bar-and-joint mechanism) and therefore has no "configurations".
However, it is possible to associate with it a cell complex which is proven to be
a combinatorial manifold.

Finally, for the sake of completeness, we discuss in Sect. 4 the cell
complex for the case when the manifold $ M(L)$ is singular.

The complex $ \mathcal{K}(L)$ already appeared in [Kapovich and Millson1995] in a slight disguise,
where it was mentioned as a "tiling of $ M(L)$". Moreover, based on the
Deligne-Mostow map, Kapovich and Millson deduced that $ \mathcal{K}(L)$ can be
realized as a piecewise linear manifold in the hyperbolic space.

We start with necessary preliminaries.
Convex configurations
A configuration $ P$ is convex
if (1) it is a convex (piecewise linear) curve, (2) no two consecutive edges are
collinear, and (3) the orientation induced by the numbering goes
counterclockwise.

The set of all convex configurations we denote by $ M_{conv}(L)$. The
set $ \overline{M}_{conv}(L)$ is the closure of $ M_{conv}(L)$ in $ M(L)$.

Lemma
1.2.

(M. Kapovich, Personal communications 2013)

(1) The set
$ M_{conv}(L)$ is an open subset of $ M(L)$ homeomorphic to the open
$ (n-3)$-dimensional ball.

(2) The closure $ \overline{M}_{conv}(L)$ is homeomorphic to the
closed $ (n-3)$-dimensional ball.

(3) The interior of $ \overline{M}_{conv}(L)$ coincides with
$ M_{conv}(L)$.

Proof.

Following paper [Kapovich and Millson1995], consider configurations
of $ n$ (not necessarily all distinct) points $ p_i$ in the real
projective line $ \mathbb{R}P^1$, which we identify with $ S^1$. Each point
$ p_i$ is assigned the weight $ l_i$. The configuration of
(weighted) points is called stable
if sum of the weights of coiciding points is less than half the weight of all
points.

The group $ PSL(2,\mathbb{R})$ naturally acts on the space of configurations.
A remarkable fact is that the quotient space of stable configurations is exactly
the space $ M(L)$. More detailed, take a stable configuration
$ \{p_i\}$. We interpret the points $ p_i$ as unit vectors in
$ \mathbb{R}^2$. In the orbit of the configuration there exists a unique point (up
to rotation of $ S^1$) such that the weighted sum $ \sum l_ip_i$ is zero.
Thus each orbit gives a configuration of the linkage $ L$.

A configuration of points yields a convex polygon whenever the
numbering $ (1,\ldots,n)$ goes counterclockwise. Therefore $ M_{conv}(L)$ is
identified with the set of $ n$-tuples of counterclockwise-oriented
distinct points $ x_i$ in $ S^1=\mathbb{R}P^1$ modulo $ PSL(2,\mathbb{R})$. We can
omit the action of the group by assuming that the first three points are
$ 0,\ 1 $, and $ \infty$. The rest of the points are then given by linear
inequalities \begin{eqnarray*} 1< x_4< x_5 < \cdots < x_n < \infty , \end{eqnarray*} which implies the statement (1). The statements (2)
and (3) are now straightforward. ⬜

Polytopes
We shall use the combinatorial structure of the following polytopes:

The permutohedron $ \Pi_n$ (see [Ziegler1995]) is defined as the convex hull of all
points in $ \mathbb{R}^n$ that are obtained by permuting the coordinates of the
point $ (1,2,\ldots,n)$. It has the following properties:

(1) $ \Pi_n$ is
an $ (n-1)$-dimensional polytope.

(2) The $ k$-dimensional faces of $ \Pi_n$
are labeled by ordered partitions of the set $ \{1,2,\ldots,n\}$ into $ (n-k)$
non-empty parts. In particular, the vertices are labeled by the elements of the
symmetry group $ S_n$. The label of a vertex is obtained by inverting
the permutation of the coordinates of the vertex.

(3) A face $ F'$ of
$ \Pi_n$ is contained in a face $ F$ iff the label of $ F'$ is
finer than the label of $ F$. Here by a
refinement
of an ordered partition $ \lambda$ we mean an ordered refinement
$ \lambda'$ whose ordering is inherited from $ \lambda$. For instance,
$ \{1,3\}\{2,4\}\{5\}$ refines $ \{1,3\}\{2,4,5\}$ and does not refine $ \{2,4,5\}\{1,3\}$.

(4) A face of
$ \Pi_n$ is the Cartesian product of permutohedra of smaller dimensions.

(5) The
permutohedron is a zonotope
, that is, the Minkowski sum of line segments.

(6) The permutohedra $ \Pi_1$,
$ \Pi_2$, and $ \Pi_3$ are a one-point polytope, a segment, and a
regular hexagon, respectively. The permutohedron $ \Pi_4$ (with its
vertices labeled) is depicted in Fig. 1.

Figure 1 Permutohedron
$ \Pi_4$.

The cyclic polytope $ C(d,n)$ is the convex hull of $ n$
distinct points $ x_1,\ldots,x_n$ on the moment curve in $ \mathbb{R}^d$, see
[Ziegler1995]. Its combinatorics is completely
defined by the following property ( Gale evenness
condition
): a $ d$-subset $ F \subset \{x_1,\ldots,x_n\}$ forms a facet of $ C(d,n)$ iff any
two elements of $ \{x_1,\ldots,x_n\} {\setminus} F$ are separated by an even number of elements
from $ F$ in the sequence $ x_1,\ldots,x_n$.

2. The Complex $ \mathcal{K}(L)$

Labeling the polygons
To explain the cell decomposition of the moduli space, we associate labels to its
points.

Assume first that a configuration $ P=(p_1,\ldots,p_n)\in M(L)$ has no parallel edges,
that is, no edgevectors $ \overrightarrow{p_ip}_{i+1}$ and $ \overrightarrow{p_jp}_{j+1}$ are parallel and
codirected.

Then there exists a unique convex polygon $ \overline{P}$ such that

(1) The edges of
$ P$ are in one-to-one correspondence with the edges of
$ \overline{P}$. The bijection preserves the directions of the vectors.

(2) The orientations of
the edges of $ \overline{P}$ give the counterclockwise orientation of
$ \overline{P}$.

In other words, the edges of $ \overline{P}$ are the edges of
$ P$ coming in the order of their slopes (see Fig. 2).
Obviously, $ \overline{P} \in M_{conv}(\lambda L)$ for some permutation $ \lambda \in S_n$. The
permutation is defined up to the action of the group generated by the cyclic
permutation $ (2,3,4,\ldots,n,1)$. The orbit of a permutation under the action of
the group is a cyclic ordering on the set $ [n]$. Summarizing the
above, our construction assigns to $ P$ the label $ \lambda(P)$ which
is a cyclic ordering on the set $ [n]$. Equivalently, expecting further
discussion on polygons with parallel edges, we state that a label of a
configuration without parallel edges is a cyclically ordered partition of the set
$ [n]=\{1,2,\ldots,n\}$ into $ n$ non-empty parts.

Lemma 2.1.

Given a cyclically ordered partition
$ \lambda$ of the set $ [n]$ into $ n$ non-empty parts,
the subset of $ M(L)$ of all polygons labeled by $ \lambda$ is an open
$ (n-3)$-ball.

Proof.

The
rearranging construction maps the set of polygons labeled by $ \lambda$
bijectively to $ M_{conv}(\lambda L)$, which is a ball by Lemma 1.2.
⬜

Definition 2.2.

Farber and
Schütz
([Farber and Schütz2007]) A set
$ I\subset [n]=\{1,2,\ldots,n\}$ is called short
, if \begin{eqnarray*} \sum_{I}^{}l_i < \frac{1}{2} \sum_{i=1}^{n}l_i. \end{eqnarray*}

Definition 2.3.

A partition of the set $ [n]$ is called
admissible
if all the parts are short.

Figure 2 Labeling of a polygon with no
parallel edges.

Figure 3 Labeling of a polygon with
parallel edges.

Assume now that a configuration $ P\in M(L)$ has parallel edges. A
permutation which makes $ P$ convex is not unique. Indeed, one can
choose any ordering on the set of parallel edges. So in cooking the label, our
construction puts the indices of parallel edges in one set.

The label $ \lambda(P)$ assigned to $ P$ is a cyclically
ordered partition of the set $ [n]$, see Fig. 3 for an
example.

Lemma 2.4.

Given a cyclically ordered partition
$ \lambda$ of the set $ [n]$ into $ k$ non-empty sets, the
subset of $ M(L)$ of all polygons labeled by $ \lambda$ is either an
open $ (k-3)$-ball (if $ \lambda$ is an admissible partition), or an
empty set (if $ \lambda$ is non-admissible).

Proof.

We
apply Lemma 2.1 to the $ k$-bar linkage with frozen
together edges. Namely, we replace each collection of edges with equal slopes
by a single edge. ⬜ A remark on notation
We write a cyclically ordered partition as a (linearly ordered) string of sets
where the set containing the entry "$ n$" stands on the last position.

We stress once again that the order of the sets matters, whereas there
is no ordering inside a set. For example, \begin{eqnarray*} (\{1\} \{3 \} \{4, 2, 5,6\})\neq(\{3 \}\{1\} \{4, 2, 5,6\})= ( \{3 \}\{1\}\{ 2,4, 5,6\}). \end{eqnarray*}

Definition 2.5.

Two points from $ M(L)$ (that is, two
configurations) are equivalent
if they have one and the same label. Equivalence classes of $ M(L)$ we
call the open cells
. The closure of an open cell in $ M(L)$ is called a closed cell
. By above lemmata, all cells are homeomorphic to balls.

For a cell $ C$, either closed or open, its label $ \lambda (C)$
is defined as the label of any interior point of the cell.

Before we formulate the main theorem, remind that a CW-complex
can be constructed inductively by defining its skeleta. Once the
$ (k - 1)$-skeleton is constructed, we attach a collection of closed
$ k$-balls $ C_i$ by some continuous mappings $ \varphi_i$
from their boundaries $ \partial C_i$ to the $ (k-1)$-skeleton. For a
regular
complex, each of the mappings $ \varphi_i$ is injective, and $ \varphi_i$
maps $ \partial C_i$ to a subcomplex of the $ (k-1)$-skeleton, see [Hatcher2002]. Regularity of a complex implies that
a complex is uniquely defined by the poset of its cells. Regularity also
guarantees the existence of well-defined barycentric subdivision and (for PL
manifolds) a well-defined dual complex.

Theorem 2.6.

The above described collection of cells yields a
structure of a regular CW-complex $ \mathcal{K}(L)$ on the moduli space
$ M(L)$. Its complete combinatorial description reads as follows:

(1)
$ k$-cells of the complex $ \mathcal{K}(L)$ are labeled by cyclically
ordered admissible partitions of the set $ [n]$ into $ (k+3)$
non-empty parts.

(2) A closed cell $ C$ belongs to the
boundary of some other closed cell $ C'$ iff the partition
$ \lambda(C')$ is finer than $ \lambda(C)$.

Proof.

The
open cells are balls by Lemmata 2.1 and 2.4. The regularity of the complex follows from
Lemma 1.2,
(3). ⬜

For the complex $ \mathcal{K}(L)$ we immediately have:

Proposition 2.7.

(1) The facets of
the complex (that is, the cells of maximal dimension $ n-3$) are
labeled by cyclic orderings on the set $ [n]$.

(2) The vertices of the complex are
labeled by cyclically ordered admissible partitions of the set $ [n]$ into
three non-empty parts. In other words, they correspond to all possible
(oriented) triangles composed of segments of lengths $ l_1,\ldots,l_n$.

(3) The vertex figure
of any vertex $ v$ of the complex $ \mathcal{K}(L)$ is combinatorially
dual to the Cartesian product of three permutohedra. More precisely, the label
$ \lambda(v)$ consists of three parts. If the three parts have $ k$,
$ l$, and $ m$ elements, respectively, then the vertex figure
of $ v$ is combinatorially dual to $ \Pi_k\times \Pi_l \times \Pi_m$.

(4) The face figure of any
$ k$-dimensional face is combinatorially dual to the Cartesian
product of $ (k+3)$ permutohedra (some of these permutohedra can be
$ \Pi_1$, and thus degenerate to a point).

Proof.

The
proof follows directly from the above construction. ⬜

Example 2.8.

Let $ n=4; \ \ l_1=l_2=l_3=1,\ l_4=1/2.$ The moduli space
$ M(L)$ is known to be a disjoint union of two circles, see [Farber2008]. The cell complex $ \mathcal{K}(L)$ is
depicted in Fig. 4.

Figure 4
$ \mathcal{K}(L)$ for the 4-gonal linkage $ (1,1,1,1/2)$ .

Example 2.9.

Assume that \begin{eqnarray*} \forall i \ \ l_n+l_i> \sum_{ n\neq j\neq i} l_j. \end{eqnarray*} In this case the
moduli space $ M(L)$ is an $ (n-3)$-sphere, see [Farber2008], and the complex $ \mathcal{K}(L)$ is dual
to the boundary complex of the permutohedron $ \Pi_{n-1}$.

Proof.

Indeed, each admissible partition is of the type \begin{eqnarray*} (*, \{n\}), \end{eqnarray*} where
"$ *$" is any linearly ordered partition of $ [n-1]$ in at least
two parts. This means that the facets of $ \mathcal{K}(L)$ are in a natural
bijection with the vertices of $ \Pi_{n-1}$. It remains to observe that the
patching rules for $ \mathcal{K}(L)$ are exactly dual to those of the
permutohedron. ⬜

In a regular complex, the boundary of each cell is a combinatorial
sphere, so it makes sense to speak of combinatorics of a cell. Let us look what
types of combinatorics do we encounter in complexes $ \mathcal{K}(L)$ for different
linkages $ L$.

Example 2.10.

Let $ n=5$, $ L=(1,1,1,1,1)$. Then
$ \mathcal{K}(L)$ is a surface of genus four patched of 24 pentagons. Each vertex
has $ 4$ incident edges. The complex is
flag-transitive
, which means that any combinatorial equivalence of any two pentagons
extends to an automorphism of the entire complex.

Example 2.11.

However, unlike the previous example, the complex is not completely
transitive, just facet-transitive
: for every two facets there exists an automorphism of $ \mathcal{K}(L)$ mapping
one facet to the other.

Proof.

Fix
a facet $ C$ of $ \mathcal{K}(L)$. Without loss of generity we may
assume that its label is $ (\{1\}\{2\}\{3\}\{4\}\ldots\{n\})$. Consider the following "starlike"
bijection $ \varphi$ which maps the vertices $ x_1,\ldots,x_n$ of the cyclic
polytope $ C(n-3,n)$ to facets of the cell $ C$: \begin{eqnarray*} &&\varphi(x_{2i+1})=(\{1\}\{2\}\{i+1,i+2\}\{3\}\{4\}\ldots\{n\}),\\ &&\varphi(x_{2i})=(\{1\}\{2\}\{k+i-1,k+i\}\{3\}\{4\}\ldots\{n\}). \end{eqnarray*}
Informally, the defining rule of $ \varphi$ is the way of drawing a
polygonal star (say, a pentagram). It is easy to check that $ \varphi$
yields a combinatorial duality. ⬜

Proposition 2.12.

(1) Faces of
$ \mathcal{K}(L)$ are combinatorially equivalent to
convex polytopes
Let $ C$ be a closed cell of $ \mathcal{K}(L)$ for some polygonal linkage
$ L$. The boundary complex of $ C$ is combinatorially
equivalent to a simple $ k$-polytope with at most $ k+3$
facets. Moreover, there exists some even $ D\in \mathbb{N}$ such that the
boundary complex of $ C$ is combinatorially equivalent to a face of
the dual to the cyclic polytope $ C(D,D+3)$.

(2) Universality
property
Conversely, any simple $ k$-dimensional polytope $ K$
with at most $ k+3$ facets arises in this way. That is, there exist a
number $ n$, an $ n$-linkage $ L$, and a cell
$ C$ of the complex $ \mathcal{K}(L)$ such that the boundary complex
of $ C$ is combinatorially equivalent to $ K$.

Proof.

(1) We may
assume that all $ l_i$ are integers, and that their sum $ D+3=\sum l_i$
is odd. Indeed, neither a small perturbation nor a scaling changes the
combinatorics of the complex. The space $ M(L)$ embeds in a natural
way in the moduli space of the equilateral polygon with $ D+3$ edges
$ M(\underbrace{1,1,\ldots,1}_{D+3}).$ The embedding maps a polygon with edgelengths $ l_1,\ldots,l_n$
to the equilateral polygon which represents the same curve, that is, with first
$ l_1$ edges parallel, next $ l_2$ edges parallel, etc. The
embedding respects the structure of cell complexes, and therefore, realizes
$ \mathcal{K}(L)$ as a subcomplex of the complex $ \mathcal{K}(\underbrace{1,1,\ldots,1}_{D+3})$, whose facets are
combinatorial cyclic polytopes (see Example 2.11).

(2) Assume that a
simple $ k$-dimensional polytope $ K$ has $ k+3$
facets. Then the dual polytope $ K^*$ has $ k+3$ vertices. We
shall prove that every simplicial $ k$-polytope with at most
$ k+3$ vertices is a face figure of the cyclic polytope $ C(D,D+3)$ for
some even $ D$. The Gale diagram of $ K^*$ (see [Ziegler1995]) is a one-dimensional configuration of
distinct black and white points. Remind that the Gale diagram of
$ C(D,D+3)$ is the alternating configuration of distinct black and white
points in the straight line. Being translated to the Gale diagram's language,
the statement we need reads as "any configuration of distinct black and white
points in the straight line can be completed to an alternating configuration of
distinct black and white points", which is obvious. If $ K$ has less
than $ k+3$ facets, the proof is even simpler.

⬜

3.
The Dual Complex $ \mathcal{K}^*(L)$: Surgery on the Permutohedron

Theorem 3.1.

The dual cell complex $ \mathcal{K}^*(L)$ carries a
natural structure of a polyhedron.

Proof.

The
cells of the dual complex $ \mathcal{K}^*$ are the duals to the face figures of
$ \mathcal{K}(L)$. By Theorem 2.6, the latter are combinatorially equivalent to
Cartesian products of permutohedra. To realize $ \mathcal{K}^*$ as a
polyhedron, for each facet of $ \mathcal{K}^*$ we take the Cartesian product of
three standard permutohedra. Their faces that are identified via isometries.
⬜

We describe below a realization of $ \mathcal{K}^*$ in the Euclidean
space $ \mathbb{R}^{n-2}$. For this, we need a preliminary construction which is the
subject of paper [Panina2015]. The construction involves the theory
of virtual polytopes
developed originally in [Pukhlikov and Khovanskii1993], and some related
technique. For the very first orientation we recommend the reader just to trust
that there exists a well-defined Minkowski
subtraction
of convex polytopes, and that Minkowski differences have a well-defined facial
structure. For more details, we refer to the above mentioned paper.
Cyclopermutohedron
For a fixed number $ n\geq3$, we define the following regular cell complex
$ {CP}_{n}$ by listing all the closed cells together with the incidence
relations.

For $ k=0,\ldots,n-3$, the $ k$-dimensional cells
($ k$-cells, for short) of the complex $ {CP}_{n}$ are labeled by
(all possible) cyclically ordered partitions of the set $ [n]$ into
$ (n-k)$ non-empty parts.

The complex $ {CP}_{n}$ cannot be represented by a convex
polytope, since it is not a combinatorial sphere (not even a combinatorial
manifold). However, it can be represented by some virtual polytope which we
call cyclopermutohedron $ \mathcal{CP}_{n}$.

Definition 3.2.

Theorem 3.3.

[Panina2015] The poset of (proper) faces of
$ \mathcal{CP}_{n}$ is combinatorially isomorphic to the complex $ CP_{n}$.

Remark.

The sum $ S+ \sum_{i< j} q_{ij}$ equals the standard permutohedron.

In an oversimplified way, the cyclopermutohedron $ \mathcal{CP}_{n}$ can
be visualized as the permutohedron $ \Pi_{n-1}$ "with diagonals". This
means that all the proper faces of $ \Pi_{n-1}$ are also faces of
$ \mathcal{CP}_{n}$. However, $ \mathcal{CP}_{n}$ has some extra faces in comparison
with $ \Pi_{n-1}$.

For any $ n$-linkage $ L$, the complex
$ \mathcal{K}^*(L)$ automatically embeds in $ {CP}_{n}$, and therefore embeds in
the face complex of $ \mathcal{CP}_{n}$. The embedding goes as follows. Take the
permutohedron $ \Pi_{n-1}\subset \mathbb{R}^{n-1}$, assuming (as usual) that the faces of
$ \Pi_{n-1}$ are labeled by ordered partitions on the set $ [n-1]$. In
particular, the vertices of $ \Pi_{n-1}$ are labeled by permutations of the set
$ [n-1]$. We introduce the following bijection between the vertex sets
\begin{eqnarray*} \psi: Vert(\mathcal{K}^*)\rightarrow Vert(\Pi_{n-1}). \end{eqnarray*} Given a vertex of $ \mathcal{K}^*$ whose label $ \lambda$ is a
cyclically ordered set $ [n]$, the mapping $ \psi$ sends it to
the vertex of $ \Pi_{n-1}$ by cutting $ \lambda$ at the position of
"$ \{n\}$" and omitting "$ \{n\}$" from the label.

Thus, the vertices of $ \mathcal{K}^*$ are geometrically realized by
vertices of the permutohedron. Next, we realize the cells of the complex: take
a cell $ C$ and patch the face of the cyclopermutohedron which
corresponds to $ C$ by Theorem 3.3.

This construction can be reformulated as the following surgery
algorithm:

(1) Start with the
complex $ \mathcal{K}^*(L)$ and the boundary complex of the permutohedron
$ \Pi_{n-1}$. Realize the vertices of $ \mathcal{K}^*$ as the vertices of
$ \Pi_{n-1}$ via the above described mapping $ \psi$.

(2) For every face
$ F$ of $ \Pi_{n-1}$ do the following. The face is labeled by some
$ \lambda$, which is a linearly ordered partition of $ \{1,\ldots,n-1\}$. If the
partition is admissible (that is, all the parts are short), keep the face
$ F$ and assign to it the label $ (\lambda, \{n\})$. If the partition is not
admissible, remove the face $ F$ from the complex. This step gives a
realization of all the cells of $ \mathcal{K}^*$ whose label contains the
one-element set $ \{n\}$.

(3) Take all the cells $ C$ of $ \mathcal{K}^*$
such that the part of $ \lambda(C)$ containing $ n$ has more than
one element. Patch in the corresponding face of the cyclopermutohedron,
which up to a translation equals \begin{eqnarray*} \sum q_{ij} - \sum r_i, \end{eqnarray*} where the first (Minkowski) sum
extends over all $ i< j< n$ such that $ i$ and $ j$
belong to one and the same part of the partition $ \lambda(C)$, and the
second sum extends over all $ i< n$ such that $ i$ and
$ n$ belong to one and the same part of the partition $ \lambda(C)$.
This is a virtual polytope with the vertex set $ \psi (Vert(C))$.

Example 3.4.

Let $ L$ be as in Example 2.9. The
above described surgery leaves the permutohedron as it is. That is, all the
faces of $ \ \Pi_{n-1}$ survive on the second step of the surgery algorithm, and
nothing is added on the third step.

Important is that the "long" edge is the last one. Otherwise we would
get another surgery, but, of course, an isomorphic combinatorics.

Example 3.5.

Let $ n=5$; $ l_1=1,2;\ l_2=1;\ l_3=1;\ l_4=0,8;\ l_5=2,2$. The
surgery algorithm starts with the permutohedron $ \Pi_4$ (see Fig.
5). The two
shadowed faces are labeled by $ (\{123\}\{4\})$ and $ (\{4\}\{123\})$. Since the
partitions $ (\{123\}\{4\}\{5\})$ and $ (\{4\}\{123\}\{5\})$ are non-admissible, according to
the algorithm, the faces are removed. All other faces of the permutohedron
survive the surgery. Step 3 gives six new "diagonal" rectangular faces. They
correspond to the cells labeled by $ (\{1\}\{2\}\{3\}\{45\})$, $ (\{1\}\{3\}\{2\}\{45\})$, $ (\{2\}\{1\}\{3\}\{45\})$,
$ (\{2\}\{3\}\{1\}\{45\})$, $ (\{3\}\{1\}\{2\}\{45\})$, and $ (\{3\}\{2\}\{1\}\{45\})$.

Figure 5 The
complex $ \mathcal{K}^*(L)$ for the $ 5$-linkage $ L=(1,2;\ 1;\ 1;\ 0,8;\ 2,2)$. We
remove from the permutohedron the grey facets and patch in the blue cylinder
.

Example 3.6.

Let $ n=5$, $ L=(3,\ 1,\ 1,\ 4,\ 4)$. Figure
6 presents the
permutohedron, the labels of the vertices, and the coordinates of the vertices
(in bold). We also depict the hexagonal face labeled by $ (\{1\}\{4\}\{235\})$. It is
the Minkowski sum of two negatively weighted and one positively weighted
segments.

Figure 6 A
"diagonal" face .

For more examples of the surgery see [Gorodetskaya2017], where Gorodetskaya presented
the surgery for all types of five-linkages.

4. Concluding Remarks

The construction of $ \mathcal{K}$ and $ \mathcal{K}^*$ suggests some
further natural discussions sketched briefly in this section.
Quasilinkages, Simple Games, Alexander
Self-Dual Complexes, and Associated Manifolds
An elementary observation is that the complex $ \mathcal{K}(L)$ depends only on
the collection of admissible partitions. In turn, these are defined by the
collection of short sets. This suggests the following generalization, which is
described in details in [Galashin and Panina2016], and which we sketch
very briefly now.

Definition 4.1.

A family $ \mathcal{F}$ of subsets of
$ [n]$ is called a quasilinkage
, if it satisfies the following properties:

The proposed notion exists in the literature; yet in completely different
frameworks. It appeared as "simple game with constant sum" in game theory,
as "strongly complementary simplicial complex", and as "Alexander self-dual
simplicial complex".

Being motivated by polygonal linkages, we call any $ S\in \mathcal{F}$
a short set
, and any $ S\notin \mathcal{F}$ a long set
.

Each polygonal linkage $ L$ yields a collection of short sets,
and therefore, is a quasilinkage. The converse is not true: there exist many
quasilinkages that cannot be represented by length assignments.

We associate with a quasilinkage $ \mathcal{F}$ a cell complex
$ \mathcal{K}(\mathcal{F})$ by applying the rules from Theorem 2.6. In [Galashin and Panina2016] it is proven that the
complex is a (combinatorial) manifold of dimension $ (n-3)$ which is
locally isomorphic to $ \mathcal{K}(L)$ for some linkage $ L$ (however,
$ L$ depends on the location, and there may be no linkage associated
to the entire complex).
Cell Decomposition for Singular Configuration
Spaces
A similar cell complex exists also for singular configuration spaces, that is, for
the case when $ L$ has configurations that fit in a straight line.

Definition 4.2.

For a singular case, a partition of
$ L=(l_1,\dots ,l_n)$ is called admissible
if one of the two conditions holds:

(1) The number of
the parts is greater than $ 2$, and the total length of any part is
strictly greater than the total length of the rest.

(2) The number of parts equals $ 2$,
and the lengths of the parts are equal.

The combinatorics of the complex $ \mathcal{K}(L)$ is literally the same
as in Theorem 2.6 except for the following items:

(1) Non-singular
vertices are labeled by admissible partitions with exactly three parts.

(2) Singular vertices
are labeled by admissible partitions with exactly two parts.

(3) Assume that a singular vertex
$ v$ of $ \mathcal{K}(L)$ corresponds to an ordered partition of
$ \{1,2,\ldots,n\}$ into two non-empty parts, say, with $ k$ and
$ l$ elements. Then the vertex figure of $ v$ is
combinatorially equivalent to the cone over $ (\partial \Pi_k\times \partial \Pi_l)^* $.

Acknowledgements.

I am grateful to Nikolai Mnev
for inspiring conversations. I am also indebted to Misha Kapovich for
delivering me the proof of Lemma 1.2.